> Did the solution taking 10 minutes make it seem like it was all just semantics from old faulty definitions?
I recall that I, and the rest of the class, were very suspicious of the proof. The proof took maybe 10 minutes, but it probably took another 10-15 minutes for the professor to convince us there wasn't a logical error in the given proof. Though the situation was kind of the opposite of what you would thing. We understood field extensions and the symmetries of the roots of polynomials really well. What took convincing was that any formula using addition, subtraction, multiplication, division, exponents, and rational roots would always give you a field extension that mapped to a "solvable group". The proof is essentially:
1. Any field extension of a number constructed using those mathematical operations must map to a solvable group.
2. For every group there exists a corresponding field extension (this is a consequence of the fundamental theorem of Galois theory).
3. There exist groups that are not solvable.
4. Therefore, there are polynomials with roots that can't be constructed from the elementary mathematical operations.
Basically the entire course is dedicated to laying out part 3, and the part we were suspicious about was part 1.
The one thing that is interesting about the proof is that it is actually partially constructive. Because there is no general quintic formula, but there are some quintics that are solvable. For instance, x^5 - 1 clearly has root x=1. And Galois theory allows you to tell the difference between those that are solvable and those that are not. It allows you to take any polynomial and calculate the group of symmetries of those roots. If that group is solvable, then all of the roots can be defined in terms of elementary operations. If not, at least one of the roots cannot.
> How do you personally imagine trisecting an angle now? Is it possible to describe your new intuition of the impossibility in different human understandable terms that are also geometric?
So the trisection proof I don't remember as well, but looking it up it isn't very geometric. It essentially proves that trisecting an angle with a compass and straight edge is equivalent to solving certain polynomial equations with certain operations, and goes into algebra.
That said, Galois theory itself feels very "geometric" in the roughest sense of the term. Fundamentally, it's about classifying the symmetries of an object.
The reason that the proof isn't geometric, is that the algebriac proof is a proof that Euclidean geometry is incomplete. How can you use a language (any given language!) to express the idea that the selfsame language is incapable of expressing a certain concept?
You can draw a picture of trisecting an angle using an ruler (with cube-root markings) or an Archimedian sprial, which are clearly more powerful than purely Euclidean geometery, but how can you draw a picture of it being impossible without something like this?
How do you draw a picture of something that doesn't exist?
2. For every group there exists a corresponding field extension (this is a consequence of the fundamental theorem of Galois theory).
Just a nit, but when talking about extensions of Q, this is called the Inverse Galois Problem and it is still an open problem.
That said, you don’t actually need this strong of a statement to show general insolvability of the quintic. Rather you just need to exhibit a single extension of Q with non-solvable Galois group. I believe adjoining the roots of something like x^5+x+2 suffices.
Your story makes me picture Geometric Algebra, defining Complex Numbers or Quaternions and multiplication on them, and their symmetries and elegant combinations. Ty for sharing your math memories. Makes me wonder if Galois theory can determine valid or invalid imaginary number combinations / systems.
I recall that I, and the rest of the class, were very suspicious of the proof. The proof took maybe 10 minutes, but it probably took another 10-15 minutes for the professor to convince us there wasn't a logical error in the given proof. Though the situation was kind of the opposite of what you would thing. We understood field extensions and the symmetries of the roots of polynomials really well. What took convincing was that any formula using addition, subtraction, multiplication, division, exponents, and rational roots would always give you a field extension that mapped to a "solvable group". The proof is essentially:
1. Any field extension of a number constructed using those mathematical operations must map to a solvable group. 2. For every group there exists a corresponding field extension (this is a consequence of the fundamental theorem of Galois theory). 3. There exist groups that are not solvable. 4. Therefore, there are polynomials with roots that can't be constructed from the elementary mathematical operations.
Basically the entire course is dedicated to laying out part 3, and the part we were suspicious about was part 1.
The one thing that is interesting about the proof is that it is actually partially constructive. Because there is no general quintic formula, but there are some quintics that are solvable. For instance, x^5 - 1 clearly has root x=1. And Galois theory allows you to tell the difference between those that are solvable and those that are not. It allows you to take any polynomial and calculate the group of symmetries of those roots. If that group is solvable, then all of the roots can be defined in terms of elementary operations. If not, at least one of the roots cannot.
> How do you personally imagine trisecting an angle now? Is it possible to describe your new intuition of the impossibility in different human understandable terms that are also geometric?
So the trisection proof I don't remember as well, but looking it up it isn't very geometric. It essentially proves that trisecting an angle with a compass and straight edge is equivalent to solving certain polynomial equations with certain operations, and goes into algebra.
That said, Galois theory itself feels very "geometric" in the roughest sense of the term. Fundamentally, it's about classifying the symmetries of an object.